Yes, but the citizen need the approval and permission to travel from the local authorities, the so called "exit visa" or Carta Blanca (White Card)
Update February 2013
The "exit visa" is no longer needed, but the authorities still may refuse to a citizen the right to travel abroad. Note: A foreign visa from the inviting country and a passport is still needed.
The procedure to leave Cuba is a multi step process:
1) Apply for a "letter of invitation". This is a document provided by a Cuban notary stating
the identity of the foreign invitor and the identity of the invited citizen.
This document is signed by the three parties, the citizen, the foreign invitor,the authorities in Cuba.
By signing the document the invitor accept to bear full responsability and to pay all costs and support for the Cuban citizen, including cost of travel, lodging, medical treatment etc.
This document costs 140 cuc.
2) Apply for a Cuban passport
3) Pass a medical examination
4) Apply for a foreign visa, from the country of destination. This can be the hard part,
because for many countries of destination, the Cuban citizen has to prove his solvency and resources. This might be easier when he or she is married to a foreigner.
In most cases the inviting foreigner too has to prove his solvency and has to sign a statement that he has the means to support the invited Cuban.
5) Apply for the "exit visa" (Carta Blanca)
The 'exit visa' is a permission of the local authorities to leave the country and travel abroad.
To get this Cuban exit visa can take some time depending on several factors:
See Update above
If the citizens are professionals (doctor, nurse, teacher, professor, scientist, etc.)
the visa can only be obtained after 3 years or even not granted.
If the Cuban has a criminal record, he or she might only leave the country 10 years after the facts.
Political involvement in anti-government actions or disidents may not get a permission.
Sometimes the permission can be refused without a clear reason or declaration.
Other ways to leave the island:
For some people the exit visa is hard to get, so they try to leave Cuba illegal by raft or boat the so called "balseros". No need to say that this is a very dangerous operation and several immigrants are drowned.
Other Cubans try to leave through a "fake marriage", or marriage fraud to get their exit visa.
The Mariel Boat Lift - Cuban Emigration
In the late 1970’s the economy on the island suffered a severe economic crisis. During this time, it became evident that the majority of people on the island were living in extreme poverty, and there was very little being done by the government. During this time, Fidel Castro even released a statement saying that anyone was free to leave if they did not like where they lived.
Between the months of April and October of 1980, many Cuban people took this suggestion. Setting sail from the Mariel Harbor in Cuba, more than a hundred thousand Cuban immigrants sailed to the southern area of Florida. However, many of the Cubans who fled the island were convicted criminals, leading to a marked rise in crime in the United States.
Note: the 1980 peak is the Mariel Exodus, after this peak the emigration stays at a constant low level till 1995.
The Modern Day Cuban Immigration
After 1980, it became evident that the United States government needed stricter immigration rules. From that time, more laws were put into effect, and the Coast Guard began regular patrols of the waters between Florida and Cuba. This slowed the number of illegal immigrants seeking to gain access to the country, but it did not stop it.
Even in modern times, Cubans still seek to avoid the Coast Guard and reach American soil. The boat people have become so well known that they have even been featured in a Hollywood documentary. This film, called “The Balseros,” is in reference for the Spanish term for "rafters". The documentary reveals how Cubans seek immigration into the United States, often floating the ocean on small rafts.
Cuban Population in America
Since Florida is so close to Cuba geographically, it is no surprise that some of the highest concentrations of Cuban immigrants can be found in the southern area of the state, specifically Miami. However, there are concentrated populations of the immigrants in other areas of the country as well.
Perhaps the most well known of these areas is Little Havana Miami. This sector of the Florida city is even counted on the United States Census as a separate area in population and culture. Because the large amount of Cuban immigrants, Little Havana has become a city so to speak with a strictly Cuban population.
Other areas of the United States that have a large Cuban concentration include Tampa Bay, Florida, North Jersey and Western areas of New York City.
Because the island has a history of independence issues, poverty, battles, and dictatorship, immigration to the United states has been a continued issue for hundreds of years. Through newly enforced laws, some of this has slowed, but boat people continue to seek access to the country.